The Knothe-Rosenblatt distance and its induced topology

TL;DR

This paper introduces the Knothe-Rosenblatt distance, a new metric on probability measures in ^N, showing it metrizes the adapted weak topology, is a geodesic distance, and extends to multi-dimensional cases.

Contribution

It establishes the Knothe-Rosenblatt distance as a new metric that metrizes the adapted weak topology and proves its geodesic properties, extending previous results to higher dimensions.

Findings

The Knothe-Rosenblatt distance metrizes the adapted weak topology.

It is a geodesic distance on ^N.

The results extend to multi-dimensional probability measures.

Abstract

A basic and natural coupling between two probabilities on $\mathbb R^N$ is given by the Knothe-Rosenblatt coupling. It represents a multiperiod extension of the quantile coupling and is simple to calculate numerically. We consider the distance on $\mathcal P (\mathbb R^N)$ that is induced by considering the transport costs associated to the Knothe-Rosenblatt coupling. We show that this Knothe-Rosenblatt distance metrizes the adapted weak topology which is a stochastic process version of the usual weak topology and plays an important role, e.g. concerning questions on stability of stochastic control and probabilistic operations. We also establish that the Knothe-Rosenblatt distance is a geodesic distance, give a Skorokhod representation theorem for the adapted weak topology, and provide multi-dimensional versions of our results.